lcm of 42 and 12
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 42 and 12. To find this, we need to identify the smallest positive integer that both 42 and 12 can divide without leaving a remainder.
Answer
The LCM of 42 and 12 is 84.
Answer for screen readers
The least common multiple (LCM) of 42 and 12 is 84.
Steps to Solve
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Find the Prime Factorization of Each Number
Start by breaking down the numbers 42 and 12 into their prime factors.
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For 42: $$ 42 = 2 \times 3 \times 7 $$
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For 12: $$ 12 = 2^2 \times 3 $$
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Identify the Highest Powers of Each Prime Factor
Next, we look at each prime factor found in the factorizations and take the highest power of each.
- For the prime factor 2: The highest power is $2^2$ (from 12).
- For the prime factor 3: The highest power is $3^1$ (from both).
- For the prime factor 7: The highest power is $7^1$ (from 42).
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Multiply the Highest Powers Together
Now, we multiply these highest powers to get the LCM.
$$ \text{LCM} = 2^2 \times 3^1 \times 7^1 $$
Calculate the product:
- First, $2^2 = 4$.
- Then, $4 \times 3 = 12$.
- Finally, $12 \times 7 = 84$.
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Final Result
Thus, the least common multiple of 42 and 12 is 84.
The least common multiple (LCM) of 42 and 12 is 84.
More Information
The least common multiple (LCM) is useful in many areas of math, particularly in solving problems that involve adding or subtracting fractions, where a common denominator is needed.
Tips
- Forgetting to consider the highest powers of the prime factors: Sometimes, it's easy to overlook taking the maximum exponent for the prime factors, especially if the factors are repeated in different numbers.
- Miscalculating the multiplication of the prime factors when combining them to find the LCM.
